"The function of a mathematician is to do something, to prove new
theorems, to add to mathematics, and not to talk about what he or
other mathematicians have done."
"If a man is in any sense a real mathematician, then it is a hundred
to one that his mathematics will be better than anything else he can
do, and that he would be silly if he surrendered any decent
opportunity of exercising his one talent in order to do
undistinguished work in other fields."
"No mathematician should ever allow himself to forget that
mathematics, more than any other art or science, is a young man's
game....Galois died at twenty-one, Abel at twenty-seven, Ramanujan at
thirty-three, Riemann at forty. There have been men who have done
great work a good deal later; Gauss's great memoir on differential
geometry was published when he was fifty (though he had had the
fundamental ideas ten years before). I do not know an instance of a
major mathematical advance initiated by a man past fifty....It is very
hard to find an instance of a first-rate mathematician who has
abandoned mathematics and attained first-rate distinction in any other
" 'Immortality' may be a silly word, but probably a mathematician has
the best chance of whatever it may mean."
"A MATHEMATICIAN, like a painter or a poet, is a maker of patterns. If
his patterns are more permanent than theirs, it is because they are
made with ideas. ... The mathematician's patterns, like the painter's
or the poet's, must be beautiful; the ideas, like the colours or the
words, must fit together in a harmonious way. Beauty is the first
test: there is no permanent place in the world for ugly mathematics."
"The best mathematics is serious as well as beautiful--'important' if
you like, but the word is very ambiguous, and 'serious' expresses what
I mean much better."
"It is undeniable that a good deal of elementary mathematics-- and I
use the word 'elementary' in the sense in which professional
mathematicians use it, in which it includes, for example, a fair
working knowledge of the differential and integral calculus) has
considerable practical utility. These parts of mathematics are, on the
whole, rather dull; they are the parts which have the least aesthetic
value. The 'real' mathematics of the 'real' mathematicians, the
mathematics of Fermat and Euler and Gauss and Abel and Riemann, is
almost wholly 'useless'(and this is as true of 'applied' as of 'pure'
mathematics. It is not possible to justify the life of any genuine
professional mathematician on the ground of the 'utility' of his
G.H. Hardy, A Mathematician's Apology